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1. Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets

   https://doi.org/10.1007/s00041-024-10093-z  

   Daskalakis, Leonidas (June 2024, Journal of Fourier Analysis and Applications)

   Abstract We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic setsB. As a corollary we obtain the corresponding pointwise convergence result on$$L^1$$ $L^1$. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on$$L^1$$ $L^1$of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund alongBon$$L^p$$ $L^p$,$$p>1$$ $p>1$, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates. 

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2. Regularity of area minimizing currents mod p

   https://doi.org/10.1007/s00039-020-00546-0  

   De Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Stuvard, Salvatore (October 2020, Geometric and Functional Analysis)

   Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure. 

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3. From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

   https://doi.org/10.1007/s00029-024-00932-8  

   Lauda, Aaron D; Licata, Anthony M; Manion, Andrew (July 2024, Selecta Mathematica)

   Abstract We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category$$\mathcal {O}$$ $O$of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams on sign variation and the combinatorics of the$$m=1$$ $m=1$amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of$$\mathfrak {gl}(1|1)$$ $\mathrm{gl}\left(1\right|1)$, and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement. 

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4. The duality resolution at $n=p=2$

   https://doi.org/10.1007/s00209-025-03754-2  

   Beaudry, Agnès; Bobkova, Irina; Henn, Hans-Werner (July 2025, Mathematische Zeitschrift)

   Abstract Working at the prime 2 and chromatic height 2, we construct a finite resolution of the homotopy fixed points of MoravaE-theory with respect to the subgroup$$\mathbb {G}_2^1$$ $G_2^1$of the Morava stabilizer group. This is an upgrade of the finite resolution of the homotopy fixed points ofE-theory with respect to the subgroup$$\mathbb {S}_2^1$$ $S_2^1$constructed in work of Goerss–Henn–Mahowald–Rezk, Beaudry and Bobkova–Goerss. 

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5. Temporally resolved parametric assessment of Z‐magnetization recovery (TOPAZ): Dynamic myocardial T ₁ mapping using a cine steady‐state look‐locker approach

   https://doi.org/10.1002/mrm.26887  

   Weingärtner, Sebastian; Shenoy, Chetan; Rieger, Benedikt; Schad, Lothar R.; Schulz‐Menger, Jeanette; Akçakaya, Mehmet (August 2017, Magnetic Resonance in Medicine)

   PurposeTo develop and evaluate a cardiac phase‐resolved myocardial T₁mapping sequence. MethodsThe proposed method for temporally resolved parametric assessment of Z‐magnetization recovery (TOPAZ) is based on contiguous fast low‐angle shot imaging readout after magnetization inversion from the pulsed steady state. Thereby, segmented k‐space data are acquired over multiple heartbeats, before reaching steady state. This results in sampling of the inversion‐recovery curve for each heart phase at multiple points separated by an R‐R interval. Joint T₁andestimation is performed for reconstruction of cardiac phase‐resolved T₁andmaps. Sequence parameters are optimized using numerical simulations. Phantom and in vivo imaging are performed to compare the proposed sequence to a spin‐echo reference and saturation pulse prepared heart rate–independent inversion‐recovery (SAPPHIRE) T₁mapping sequence in terms of accuracy and precision. ResultsIn phantom, TOPAZ T₁values with integratedcorrection are in good agreement with spin‐echo T₁values (normalized root mean square error = 4.2%) and consistent across the cardiac cycle (coefficient of variation = 1.4 ± 0.78%) and different heart rates (coefficient of variation = 1.2 ± 1.9%). In vivo imaging shows no significant difference in TOPAZ T₁times between the cardiac phases (analysis of variance:P = 0.14, coefficient of variation = 3.2 ± 0.8%), but underestimation compared with SAPPHIRE (T₁time ± precision: 1431 ± 56 ms versus 1569 ± 65 ms). In vivo precision is comparable to SAPPHIRE T₁mapping until middiastole (P > 0.07), but deteriorates in the later phases. ConclusionsThe proposed sequence allows cardiac phase‐resolved T₁mapping with integratedassessment at a temporal resolution of 40 ms. Magn Reson Med 79:2087–2100, 2018. © 2017 International Society for Magnetic Resonance in Medicine. 

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